Problem Duplicating Textures
Normally, Cycles Texture will not be defaultly linked by another material. The problem you met is because you put the texture in the node group, which can be shared as sort of "public datablock", while there is currently none for that in the Duplicate Data setting yet. Probably we should suggest some genius developer to improve this? lol

To solve this, just simply make a single user for the node group.

Finally back to your question: the "Material" and "Texture" checkbox there perfectly work for BI. As for Cycles, Material does work, but for the Cycles Texture, I don't think it's gonna work that way, since it needn't. As I said, Cycles texture is kind of "material-independent", unless being part of a node group, in other words, an independent datablock shared by materials.